Fast modelling of plasma jet and particle behaviours in spray conditions Guy Delluc, Hélène Ageorges, Bernard Pateyron, Pierre Fauchais SPCTS, UMR-CNRS 6638 University of Limoges, 123, avenue Albert Thomas 87060 Limoges Cedex, France Phone : +33(0) 55 45 74 39 e-mail :
[email protected] Abstract. This paper presents a simplified code allowing to find in a few minutes the trends of the d.c. plasma spray process, at least for a single particles or a particles in flight. It is based on a parabolic 2D flow for the plasma jet and a 3D calculation for the heat and momentum transfer to particles. It neglects the carrier gas flow rate- plasma flow interaction but the obtained trends are in good agreement with those obtained with 3D sophisticated codes. However results depend strongly on the turbulence model, the plasma effect corrections chosen for the heat and momentum transfer. Thus, as with 3D codes, the model has top be backed by experiments. It can be used to train operators and let them “see” almost immediately the effects of the different macroscopic spray parameters. The code and the plasma properties used can be freely downloaded. Key words: plasma spraying, fast modelling, particle behaviour 1
Introduction
Numerous numerical modelling for the direct current (d.c) plasma jets used for spraying have been published since the eighties (see for example [1]). However it is only very recently that the complexity of the modelling has been taken into account with 3-dimension (3D) transient models accounting for the arc root fluctuation at the anode [2,3]. These fluctuations [4], when using plasma forming gases containing diatomic gases such as H2 or N2 can induce, with the restrike mode, transient voltage V(t) fluctuations of ±25% at frequencies in 5000 Hz range, resulting for current I source in power fluctuating of ±25%. Thus the corresponding plasma jets are fluctuating in length and position. A recent calculation [2], assuming a uniform power generation in a given volume V inside the anode (V(t)*I/V) as well as experiments show that the plasma temperature
and velocity fluctuate accordingly. It is thus quite understandable that most flow models are based on the stationary behaviour of plasma jets with velocity and temperature distributions matching with the plasma gas mass flow rate and enthalpy [5, 7]. However the calculation results are strongly linked to the choice of inlet profiles, grid and turbulence model [2, 3]. As soon as particles are injected, as in the plasma spray process, the problem becomes more complex. Even the calculation of the trajectory, velocity and temperature of a single particle injected with a given injection velocity vector is not straight forward. This is due to the specific effects of the plasma [9-13]:
temperature gradient in the boundary layer surrounding the particle,
non continuum effect,
thermal buffer constituted by the vapour resulting from the particle evaporation and travelling with it because of the low Reynolds number of the latter,
the radiation emitted by the particle and principally by the metal vapour when evaporation occurs,
the heat conduction within the particle,
the turbulent dispersion of small particles.
However in the process numerous particles are simultaneously injected (for example about 108 s-1 for alumina particles 20µm in diameter, with a flow rate of 3 kg/h). Thus the three following problems have to be accounted for [13,14-21] :
The dispersion of the particles at the injector exit particle size and injection velocity vector distribution the latter being due to their collisions between themselves and the injector wall [14].
The perturbation of the plasma flow by the powder carrier gas. As emphasized by different authors [18, 8, 6, 2, 19 and 20] it plays a major role due to the interaction of the plasma flow with the cold gas which has a high momentum (the injector internal diameter being below 2 mm) and the level of turbulence generated.
The effects of the plasma jet fluctuations on the particle temperature, velocity and trajectory distributions [18, 21].
Even when neglecting the effects of the plasma jet fluctuations and using stationary models the
particle treatment should be 3D [6, 19, 20 and 22]. For example in the recent work of Remesh et al. [22], the influence of the carrier gas flow rate was studied versus the spray conditions and results confirmed by measurements. However, if the general trends given by such models are good, many parameters have to be adjusted to get a good agreement with experiments such as:
the inlet temperature and velocity radial profiles for the plasma flow
the turbulence model (of course with a low Reynolds to account for the laminar plasma core),
the specific effects of the plasma on particles,
the stochastic approach of the distribution of particle
the perturbation of the plasma jet by the cold carrier gas…
One of the main draw backs of these sophisticated codes is the computing time which is not compatible with industrial needs. That is why there is a boom for simplified models able to give quickly (in a few seconds) at least good trends. This is the goal of this paper where the most time consuming part of the model i.e. the flow modelling has been simplified while the particle injection was kept 3D. It of course, means that the influence of the cold carrier gas on the plasma jet is neglected. In the paper are presented successively:
The fast software Jets&Poudres [23] built on the GENeral MIXing (Genmix) computer code improved by using thermodynamic and transport properties closely related to the local temperature and composition of the plasma. These properties are obtained from T&TWinner database [24]. This way heat and momentum transfers to single particles are calculated and stochastic distributions are imposed to particles at the injector exit.
The results obtained for a conventional Ar-H2 d.c. plasma jet where zirconia particle are injected and the discussion of the results according to the effect of different parameters.
2
Description of the model Jets&Poudres
Various clever numerical methods were developed in the past to simulate 2-D parabolic gas flows
for laminar boundary layers or jets. For example, the Genmix algorithm developed by Spalding and Patankar [25]. It is known as the Bikini method because it requires a very low-cost memory and computing time. 2.1 Flow simulation The model Jets&Poudres, forecasts the dynamic of a single or multi particles fed in a plasma jet (see fig.1). This model, built in Visual Basic, allows a convivial exchange. Its aim is not to forecast a result very close to experiment but to: 1
compute rapidly the parameters of the plasma spray ;
2
present synthetic and explicit results;
3
give the tendencies and phenomena orders of magnitude.
Figure 1: Schematic representation of the spray parameters data used for the simulation Genmix handles the two-dimensional, parabolic flows, i.e., those of high Reynolds and Peclet numbers based on the cross stream dimension, with no recirculation. "parabolic" flows are: •
steady,
•
predominantly in one direction, defined as that in which the velocity vector has
nowhere a negative component; •
without recirculation or diffusion effects in that direction.
•
besides the conditions, the flow is assumed to be in Local Thermodynamic
Equilibrium (LTE), the carrier gas flow perturbation beeing neglected and the flow
assumed to be with no swirl. These conditions have been used as a first approximation of plasma jets. Genmix embodies a self-adaptive computational grid, which enlarges or contracts to cover only the regions of interest (hence it explains the relatively small demand on computing power and its nickname, i.e., the Bikini method). The turbulence can be simulated by different models, but in the case of a plasma jet it is the classical mixing length which appears to be the simplest and fastest. In Jets&Poudres the input data are:
the mixing length value
the mass flow rate m g0 and the composition of the plasma forming gas,
the composition of the gas atmosphere far away from the jet,
the electric current intensity I,
the electric power P with such previous conditions (obtained from experiments),
the efficiency of the energy transferred to the gas ρth
The temperature evolution of plasma properties are deduced from those of pures gases by using mixing rules and the code T&TWinner [24]. Then the specific enthalpy h = ρ th P / m g0 is calculated from the ratio of effective power ρ th P to the mass flow rate and the enthalpy temperature is obtained from the equilibrium properties at atmospheric pressure. From this temperature the specific volume of the gas is calculated and thus its mean velocity. Uniform radial profiles of temperature and velocity are assumed such as the gas enthalpy and mass flow rate at the nozzle exit are conserved. It is worth to underline that, whatever may be the profiles at the nozzle exit within a few tenth of mm the code creates its own profile independently of the starting ones.
2.2
Particle simulation
2.2.1 Behaviour of a single particle in the plasma jet For the particles sprayed by a plasma jet two types of phenomena are of interest. One is the dynamic of the movement of particles with their trajectories, velocities and accelerations. The second is their thermal history, i.e., their temperature, melting or freezing, as well as the heat flux
at their surface. a/ Momentum exchange between a single particle and the plasma jet.
Under the assumption that the Stoke’s drag is the dominant force [10, 14] in the dynamic of the particle injected in the plasma jet, the movement equation can be written as:
d 2p dv 1 mp = − C D ⋅ π ⋅ ⋅ ρ ∞ ⋅ U − v ⋅ (U − v ) + Fx dt 2 4
(1)
where: CD is the drag coefficient depending of the morphology of the particle and the Reynolds number; dp is the initial diameter of the particle (m) ; v is the particle velocity (m.s-1); U is the plasma velocity (m.s-1); ρ∞ is the plasma specific mass (kg.m-3) and µ∞ .the plasma viscosity (kg/m.s), both outside the dynamic boundary layer surrounding the particle. CD is an empirical function of the Reynolds’ number
Re =
2rρ ∞ U − v
µ∞
(2)
In this paper, according to the literature, it has been chosen [13]
(
)
⎛ 24 ⎞ C d = ⎜ ⎟ 1 + 0.11. Re 0.81 f 0 ⎝ Re ⎠
(3)
f0 is a correction factor to take into account property gradients in the boundary layer around the particle. In this paper it has been chosen ⎛ρ µ ⎞ f0 = ⎜ ∞ ∞ ⎟ ⎜ρ µ ⎟ ⎝ p p⎠
0.45
(4)
as proposed by Lee et al. [26], where subscribes ∞ and p respectively indicate plasma and particle specific mass and viscosity. The external forces Fx are rather well represented by the thermophoresis force resulting from the very high thermal gradient in the fluid and the gravity force which have been neglected here, both being low compared to the drag force. b/ Heat exchange between a single particle and the plasma jet
The heat transfer mechanisms to the particle in the plasma jet can be expressed by four successive steps [15]: the heating of the solid particle, its melting, the heating of the molten
particle and its vaporization. The governing differential equations for the temperature time evolution of a spherical particle are the following:
Solid particle heating
The particle temperature (TP), neglecting the heat propagation is calculated through the total heat energy in a film at the particle surface. Its expression is: dTP 6 ⋅ Qn = dt π ⋅ d 3p ⋅ cp ⋅ ρ P
(5)
where: Qn is the energy required for heating up the particle, it is a conduction - convection heat energy (W); cp is the specific heat at constant pressure of the particle depending of the material (J/kg.K). In this paper, to take into account the steep temperature gradients within the thermal boundary layer around the particle, the integrated thermal conductivity
~ Κ(T) =
T∞ 1 Κ(θ)dθ ∫ T∞ − 300 300
(6)
is used instead of the thermal conductivity K(T). Then with the radiative cooling it comes: ~ ~ Q n = πd 2p (T∞ − 300) Κ(T∞ ) − (Tp − 300) Κ(Tp ) - εσ S T p4 − Ta4
{
(
)}
(7)
where T∞ is the temperature outside the boundary layer, Ta is the surrounding temperature, ε is the particle emissivity (taken as 0.8) and σS the Stephan-Boltzmann constant (σS =10-9 W K-4 m2
).
Melting of the particle at constant temperature T = TF
When TP = TF (melting temperature), it is assumed that the total energy from the plasma to the particle is converted into the latent heat of fusion ∆HF. The melting mass fraction XP is governed by
dX P 6 ⋅ Qn = 3 dt π ⋅ d p ⋅ ∆H F ⋅ ρ P
(8)
where: ∆HF is the latent heat of fusion (J/kg). XP is in the range 0 to 1. If XP = 0, the particle is solid and if XP = 1, it is fully melted.
Heating of the liquid particle
Two assumptions are possible in this step: the liquid phase of the particle vaporizes or not. If the liquid phase of the particle is not vaporizing, the heating of the liquid is similar to that of the solid particle (eq. 4) with the specific heat at constant pressure of the liquid. If the liquid particle vaporizes its diameter decreases according to the following equation: d(d P ) 6 ⋅ Q 'n = dt π ⋅ d 3p ⋅ ∆H Vap ⋅ ρ P
(9)
where: ∆Hvap is the specific latent heat of particle vaporization (J.kg-1) ; Q’n is the thermal energy lost when vaporizing the particle (W). When TP = Tb (boiling temperature), the total energy from the plasma to the particle is converted in latent heat of vaporization. The diameter evolution of the particle is given by an equation similar the last one (eq. 8). 2.2.2
Behavior of a powder
To simulate the formation of a deposit, a large quantity of particles (108- 1010 /s) has to be inject within the plasma jet. Unfortunately the particles in a powder have different diameters. The particle size analysis of a commercial powder shows that they have roughly a Gaussian distribution in diameter. Thus, two cases are studied to simulate a powder, either that with the distribution given by the experimental particle size analysis or that with a Gaussian distribution according to the limit central theorem with twelve shots randomly numbered. The powder is injected by a carrier gas in the plasma jet. This phenomenon could be rapidly complex, so to simplify the model the assumptions are the following: the particles have a velocity derived from that of the carrier gas, they are not interacting between themselves, the carrier gas flow rate does not vary with time, the injector walls are smooth and straight, and the velocity of the carrier gas is not time dependent and constant. The mean injection velocity is adjusted to the mean size of the particles in such a way their trajectory makes an angle of 3.5° with the plasma jet axis. The particle collisions between themselves and with the injector wall induce a dispersion of the particle jet at the injector exit. To integrate this phenomenon, which has been measured [14] for 45+22 µm alumina particles as a cone with an angle of 20°, the model attributes to each particle an inclined angle of the exit injector velocity between 0 and 20°by firing at random a number. In order to rapidly obtain results only 32 000 particles are generated to build a sample of powder
which allows the computation of the deposit height distribution. 2.3 Coating formation The coating is constructed by taking into account only particles over their melting temperature,
those below rebounding. The particle flattening has been assumed to follow the Madjeski’s formula with disk shaped splats:
D = 1.29 Re 0p.2 d
(10)
If the impacting particles covers the more than 50% of an already deposited splat it is disposed over it, part of the new splat creating a void, while if it is covers less than 50% it is disposed aside the previously deposited splat. 3
Results and discussion
3.1 Spray conditions The spray condition are summarized in fig 1 where the torch thermal efficiency is ρth =55% for a
d.c. plasma torch PTF4 type with an anode nozzle 7mm in i.d. The powder properties (specific heat at constant pressure, specific mass, thermal conductivity, latent heat of melting and vaporization are assumed to be constant what ever may be the temperature in the solid or liquid states. 3.2
Comparison of Jets&Poudres and 3D model Estet3.4.
The plasma jet and its plume have been computed using the sophisticated code Estet3.4 [8,14] for a flow rate of 45/15 slm Ar-H2, a nozzle internal diameter of φ= 7 mm, an effective power of 21.5 kW (65 V, 600 A, ρ th = 55% ). The turbulence model of Estet was k-ε RNG and the 3D grid was 71, 88 and 71 according to the x, y, z directions. Figures 2a, 2b, 2c compare the radial profiles of velocity, temperature and surrounding atmosphere concentration at 0.0004 m, 0.0222 m, 0.0314 m, 0.0594m (thin lines) calculated both with Estet and Jets&Poudres (thick lines). It can be seen that for temperatures and velocities near the nozzle exit and far away from it profiles are very close but at the intermediate axial distance (0.0314 m) the profiles are somewhat different. In fig. 2c it can be seen that the dilution of the plume by the surrounding atmosphere is more important in the forecast from Jets&Poudres. However it should be noted that Estet underestimates the experimental results obtained with an enthalpy
probe at that distance. Fig. 2d presents the comparison of the axial profiles for temperature and it can be seen that Estet code (Estet) forecasts a smoother evolution of temperature than Jets&poudres with standard mixing length
l0
(Jgenuine) in the intermediate axial distance from the nozzle exit. However with the modified mixing length ⎛1⎞ ⎜ ⎟
⎛ T (r ) ⎞ ⎝ 9 ⎠ l m = l0 ⎜ ⎟ ⎝ 300 ⎠
(11)
the trend of Jets&Poudres is the same as that given by Estet. The modified mixing length is determine as that it is not change for room temperature and a laminar behaviour at high temperature in the plasma jet core. The power coefficient is adjusted in better agreement with experimental measurements.
3.3
Treatment of a single particle
In the section will be studied the influence, on the treatment of a single particle, of the following parameters : on the one hand mixing length and temperature gradients in the particle boundary layer for a given particle diameter and injection velocity and on the other hand the influence of the injection velocity on the particle trajectories, temperatures and velocities. 3.3.1
Mixing length
For the comparison of the flow results obtained with the Genmix and Estet3.4 code, it has already been shown in section 3.2 that to achieve a good agreement between both models along the jet axis, the Genmix mixing length lm has to be modified.
0 00
Velocity (m/s) 80 100 120 140 160 0 0 0 0 0
12
E0005 E0031 E0060
10
00
0
E0005 E0031
00
J0031
00
80
J0060
4
2
0. 01
0. 01
0. 01
8
6
0. 00
4
4 01 0.
0. 00
2 01 0.
2
6 8 01 00 00 0. 0. 0. Radial position (m)
0. 00
4 00 0.
0. 00
2 00 0.
0
0
0
0
20
00
20 0
40
40 0
00
60 0
J0060
J0005
60
Temperature (K)
E0060 J0005 J0031
Radial position (m)
0. 08
0. 07
0. 06
0. 05
Axial distance from nozzle (m)
Radial position (m)
Figure 2c
0. 04
2 4 6 8 1 2 4 6 8 2 00 00 00 00 0.0 .01 .01 .01 .01 0.0 0 0 0 0 0. 0. 0. 0.
0. 03
0
0. 02
J0031 J0060
Estet Jgenuine J&P
0. 01
0 0 0 0 0 0 0 0 0 0 .1 .2 .3 .4 .5 .6 .7 .8 .9
Percentage of surrounding
E0022 E0031 E0060 J0022
Temperature (K) 20 40 60 80 100 120 140 0 00 00 00 00 00 00 00 0
Figure 2b
1
Figure 2a
Figure 2d
Figure 2. Radial distributions of velocity (a), temperature (b) and percentage of surrounding atmosphere (c) calculated with Jets&Poudres (J curves) and Estet3.4 computer code (E curves) at different distances from the nozzles exit: E0022, J0022 at 0.0022 m, E0031, J0031 at 0.0031m, E0060, J0060 at 0.0060 m, for an Ar-H2 (45-15 slm, d.c. plasma jet, P= 36300 kW, ρ th = 0.50 , anode nozzle i.d. = 0.0007 m. (d) Comparison of the axial profile for temperature forecast by Estet code (Estet), Jets&Poudres (J) and Jets&Poudres with standard l m (Jgenuine) and Jets&Poudres with modified length (J&p)
Lm= L0
Lm= L0*1.5 Over-expanded
Lm= L0*0.5 Under-expanded
Lm= L0*(T(r)/300)1/9 Slighly under-expanded
Figure 3: Mixing length lm effect on the jet and the trajectory of a 30µm diameter zirconia particle injected with the optimum injection velocity for lm = standard value of Genmix. In fig 3 are represented the mixing length effect on the flow expansion and trajectory of a zirconia particle(d=30µm, injected in all cases with the same injection velocity adapted to achieve a trajectory making an angle of 4° with the jet axis for lm=l0) . lm as been taken as lm=l0 Genmix case reference, lm from equation (10) to a slightly underexpanded jet, lm = 0.5*l0 corresponding to an under-expanded jet and lm=1.5*l0 corresponding to an over-expanded jet. The consequences on the particle velocities are presented in Fig.4. As it could be expected the lowest velocity is obtained when the over expanded jet and in it increases drastically when the jet cone angle diminishes with the mixing length decreases.
350
Under-expanded
300
Slighly over-expanded
Velocity (m/s)
250 200 150
Genmix
100 50
Over-expanded
10 0.
0.
08
06 0.
04 0.
02 0.
0.
00
0
Axial distance from nozzle (m)
Figure 4: Mixing length lm effect on the velocity of a 30 µm diameter zirconia particle. Of course the same trend is obtained with the particle surface temperature. Such results emphasize the drastic influence of the choosen turbulence model. The same variations were obtained with the 3D Estet code [2, 8]. 3.3.2 Corrections to the drag coefficient These corrections as well as those to the heat transfer coefficient are very important. They are
illustrated in Fig 5 for the drag coefficient calculated without correction (see eqn. (3) ) and with the Lee-Pfender correction for temperature gradients. (see eqn. (4)) As it can be seen in Fig. 5, for the same injection velocity and a 30 µm diameter particle the trajectory obtained with the corrected drag coefficient is more divergent than that without correction. Of course correspondingly the velocity is lower with the corrected CD. The other corrections such as those related to vaporization, Knudsen effect (especially for particle with dp < 20µm) exhibits similar effects.
Radial distance (m )
0.
00
6
0.
00
4
0.
00
2
0.
00
0
-0 -0
.0
02
0 .0
Pfender Conventional
4
06 .0 0 -0 0 .0
0 .0
2
0 .0
4
0 .0
6
0 .0
8
0 .1
0
Axial distance from nozzle (m)
Figure 5: Influence of the correction due to the thermal gradients on the drag coefficient CD and the resulting trajectory. 3.3.3 Injection angle influence. The divergence of the particle at the injection exit (due to their collision with the injector wall
and between themselves) can be represented by injecting them at the same point with different angles (see fig. 6). This approximation is justified by the size of the injector internal diameter (
